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The Triangle and its Properties - Sum of the Lengths of Two Sides of a Triangle

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The fundamental property of any triangle is that the sum of the lengths of any two sides must be strictly greater than the length of the third side. For a triangle with sides aa, bb, and cc, we must satisfy: a+b>ca + b > c, b+c>ab + c > a, and c+a>bc + a > b.

A triangle with sides labeled a, b, and c to illustrate the triangle inequality property.
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If the sum of two sides is equal to the third side (a+b=ca + b = c), the 'triangle' collapses into a straight line, and no actual triangle is formed. If the sum is less than the third side, the ends of the two shorter sides cannot meet.

Visual representation of a + b < c where the lines do not meet to form a triangle.
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The difference between the lengths of any two sides of a triangle is always smaller than the length of the third side: ∣a−b∣<c|a - b| < c.

A triangle with sides 10, 11, and 15 showing that the difference between any two sides is less than the third.
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To determine if three given lengths can form a triangle, it is sufficient to check if the sum of the two smaller lengths is greater than the largest length.

📐Formulae

a+b>ca + b > c

b+c>ab + c > a

c+a>bc + a > b

∣a−b∣<c|a - b| < c

Difference of two sides<Third side<Sum of two sides\text{Difference of two sides} < \text{Third side} < \text{Sum of two sides}

💡Examples

Problem 1:

Check if a triangle can be formed with side lengths 8 cm, 10 cm, and 15 cm.

Solution:

Step 1: Identify the side lengths as a=8a = 8, b=10b = 10, and c=15c = 15. Step 2: Test all three sum conditions:

  1. 8+10=188 + 10 = 18. Is 18>1518 > 15? Yes.
  2. 10+15=2510 + 15 = 25. Is 25>825 > 8? Yes.
  3. 8+15=238 + 15 = 23. Is 23>1023 > 10? Yes. Since the sum of any two sides is greater than the third side in all cases, a triangle can be formed.

Explanation:

To determine if a triangle is possible, we verify the Triangle Inequality Theorem by checking if every combination of two sides added together exceeds the third side.

Problem 2:

The lengths of two sides of a triangle are 6 cm and 9 cm. Between which two numbers must the length of the third side fall?

Solution:

Step 1: Let the two given sides be s1=6s_{1} = 6 cm and s2=9s_{2} = 9 cm. Step 2: Calculate the sum of the sides: 9+6=159 + 6 = 15 cm. The third side must be less than 15 cm. Step 3: Calculate the difference of the sides: 9−6=39 - 6 = 3 cm. The third side must be greater than 3 cm. Therefore, the third side xx must satisfy 3<x<153 < x < 15.

Explanation:

The third side of a triangle is always bounded by the difference and the sum of the other two sides. Any value strictly between 3 and 15 (like 4, 7, or 14.5) would allow a triangle to be constructed.

Problem 3:

Is it possible to have a triangle with sides 33 cm, 44 cm, and 77 cm?

Diagram showing two line segments of 3cm and 4cm lying flat on a 7cm segment, failing to form a triangle.

Solution:

3+4=73 + 4 = 7 Since the sum of the two shorter sides is not greater than the third side (77 cm), a triangle cannot be formed.

Explanation:

According to the Triangle Inequality Theorem, the sum of any two sides must be greater than the third side. Here, 3+43 + 4 is equal to 77, not greater than 77.

Problem 4:

A triangle has sides of 55 cm and 1212 cm. If the third side xx is an integer, what are the minimum and maximum possible values for xx?

A triangle with sides 5, 12 and unknown side x.

Solution:

Minimum value: 12−5<x⇒7<x12 - 5 < x \Rightarrow 7 < x. The smallest integer greater than 77 is 88. Maximum value: 12+5>x⇒17>x12 + 5 > x \Rightarrow 17 > x. The largest integer less than 1717 is 1616.

Explanation:

The third side must be greater than the difference of the two sides (12−5=712 - 5 = 7) and smaller than the sum of the two sides (12+5=1712 + 5 = 17).